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REPORT 1181 



Langley Aeronautical Laboratory 
Langley Field, Va. 

National Advisory Committee for Aeronautics 

Headquarters , 1512 H Street NW., Washington 25, D. C. 

Created by act of Congress approved March 3, 1915, for the supervision and direction of the study 

Vrea e y 6 j t9 membership was increased from 12 to 15 bv act 

of the problems of flight (L. S. L ode, title ou sec ; members are appointed by the President, 

approved March 2, 1929, and to 17 by act approved May 25, 1948. 1 ne memoers ar re 

and serve as such without compensation. 

Jerome C. IIcnsaker. Sc. D., 
Detlev \V. Bronk. Ph. D.. President. 

Massachusetts Institute of Technology, Chairman 
Rockefeller Institute for Medical Research, Vice Chairman 

Joseph P. Adams, LL. D„ member. Civil Aeronautics Board. 
Allen V. Astin, Ph. D. Director. National Bureau of Standards. 
Preston R. Bassett, M. A., President. Sperry Gyroscope Co., 

Leonard Carmichael, Ph. D.. Secretary, Smithsonian Insti- 

U^LrH S Damon, D. Eng., President, Trans World Airlines Inc. 
JtMES H. Doolittle, Sc. D.. Vice President, Shell Oil Co. 
Lloyd Harrison, Rear Admiral, United States Navy, Deputy 
and Assistant Chief of the Bureau of Aeronautics. 

Ronald M. Hazen. B. S.. Director of Engineering, Allison 
Division, General Motors Corp 

Ralph A. Oestie, Vice Admiral, United States Navy, Deputy 

Chief of Naval Operations (Air). 

Donald L. Putt, Lieutenant General, United States Air Force. 

Depute Chief of Staff (Development). 

Donald ‘A. Quarles, D. Eng., Assistant Secretary of Defense 
(Research and Development). 

Arthur E. Raymond, Sc. D„ Vice President-Engineering, 
Douglas Aircraft Co., Inc. 

Francs W. Reichblderfer, So. D„ Chief, United States 
Weather Bureau. 

Oswald Ryan, LL. D. ( member, Civil Aeronautics Board. 
Nathan F. Twining, General, United States Air force. Chief 
of Staff. 

Hugh L. Dryden, Ph. D., Director 
John W. Crowley, Jr., B. S., Associate Director for Research 

John F. Victory, LL. D., Executive Secretary 
Edward II. Chamberlin, Executive Ojftcsr 

Henry J. E. Reid, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va. 

Smith J. DeFrance, D Lug., Director, Ames Aeronautical laboratory, Moffett Field, Calif. 

Edward R. Sharp, Sc. D„ Director, Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, Ohm 


Aeronautical Laboratory Ames Aeronautical Laboratory 

langley Field, Va. Moffett Field, Calif. 

Conduct, under unified control, for all agencies, of scientific research on 

Lewib Flight Propulsion Laboratory 
Cleveland Airport, Cleveland, Ohio 

the fundamental problems of flight 


REPORT 1181 


By John C. Hot bolt and Eldon E. Kordes 


An analysis is made of the structural response to gusts of 
an airplane having the degrees of freedom of vertical motion 
and wing bending flexibility and basic parameters are estab- 
lished. .4 convenient and accurate numerical solution of the 
response equations is developed for the case of discrete-gust 
encounter , an exact solution is made for the simpler case of 
continuous-sinusoidal-gust encounter , and the procedure is 
outlined for treating the more realistic condition of continuous 
random atmospheric turbulence , based on the methods of 
generalized harmonic analysis. 

Correlation studies between flight and calculated results are 
then given to evaluate the influence of wing bending flexibility 
on the structural response to gusts of two twin-engine transports 
and one four-engine bomber. It is shown that calculated 
results obtained by means of a discrete-gust approach reveal the 
general nature of the flexibility effects and lead to qualitative 
correlation with flight results. In contrast , calculations by 
means of the continuous-turbulence approach show good 
quantitative correlation with flight results and indicate a much 
greater degree of resolution of the flexibility effects. 


In the design of aircraft the condition of gust encounter has 
become critical in more and more instances, mainly because 
of increased flight speeds and because of configuration 
changes. Aircraft designers have therefore placed greater 
emphasis on obtaining more nearly applicable methods for 
predicting the stresses that develop. As a result, the number 
of papers on this subject has significantly increased. (See, 
for example, refs. 1 to 16.) Many of the papers have treated 
the airplane as a rigid body and in so doing have dealt with 
either the degree of freedom of vertical motion alone (refs. 

1 to 4) or with the degrees of freedom of vertical motion and 
pitch (refs. :j, 5, and 6). In the main, these rigid-body 
treatments tacitly involve the concept of “discrete,” “iso- 
lated” gusts, but more recently steps have been taken to 
treat the more realistic condition of continuous-turbulence 
encounter in an explicit manner (see refs. 6 to 9). 

In addition to rigid-hodv effects, one of the more impor- 
tant items that has been of concern in the consideration of 
gust penetration is the influence that wing flexibility has on 
structural response. This concern has two main aspects: 

ftt-s \ A ( ' A I N hy Irhn r. |[o:iU.h, l also muimns issmtial iimlmsi! 

}.\ I Mitii K Konl«“> mil lolin <\ llniilioli.tii.fct. 

(1) that including wing flexibility may lead to the calculation 
of higher stresses than would be obtained by rigid-body treat- 
ment of the problem and (2) that wing flexibility may intro- 
duce some error when an airplane is used as an instrument for 
measuring gust intensity. Thus, several papers have also 
appeared which treat the airplane as a flexible body. In 
most of these papers the approach used involves the develop- 
ment of the structural response in terms of the natural modes 
of vibration of the airplane (refs. 10 to 15). In others the 
approach is more unusual, as, for example, reference 16 which 
deals with the simultaneous treatment of the conditions of 
equilibrium between aerodynamic forces and structural de- 
formation at a number of points along the wing span. What- 
ever the approach, however, these flexible-body analyses have 
two main shortcomings. They too have adhered to the con- 
cept of simple-gust or discrete-gust encounter (ref. 10 is an 
exception) and also they are not very well suited for making 
trend studies without excessive computation time. 

The intent of the present report is to shed further light 
upon the case of gust penetration of an airplane having the 
degrees of freedom of vertical motion and wing bending. It 
has several objectives: (1) to establish some of the basic pa- 
rameters that are involved when wing bending flexibility is 
included, (2) to develop a method of solution which is fairly 
well suited for trend studies without excessive computation 
time, (3) to evolve methods for treating continuous turbu- 
lence as well as discrete gusts, and (4) to show the degree of 
correlation that can be obtained between flight-test and an- 
alytical results and, through this correlation, to assess how 
well flexibility effects may be analyzed. In effect, this report 
is a composite of the discrete-gust studies made jointly by 
the authors in references 11 and 12 and of the continuous- 
turbulence studies made by the first author in reference 10 
and in unpublished form. 

The report is developed as follows: The equations for re- 
sponse (including accelerations, displacements, and bending 
moments) are derived and the basic parameters outlined. A 
simple solution of these equations follows for both discrete- 
gust encounter and for continuous-sinusoidal-gust encounter. 
Next, the procedure for treating continuous atmospheric tur- 
bulence is outlined. Then, the correlation studies involving 
a comparison of flight-test results with the calculated results 
obtained for both discrete-gust and continuous-turbulence 
conditions are given. 

from NAf'A TN 2:r*:i hy John <\ lloubolt uul KMon V. Ktw it s, I*p'2. nn i \ A A i'\ 





a slope of lift curve 

a* deflection coefficient for nth mode, function of 

time alone 

A aspect ratio of wing 

b span of wing 

c chord of wing 

c 0 chord of wing midspan 

d , e , h see equation (23b) 

E Young’s modulus of elasticity 

{* S 

f(s) nondimensional gust force, J jj ^ (s — a) da 

F external applied load per unit span 

g acceleration due to gravity 

H distance to gust peak, chords 

I bending moment of inertia 

k reduced frequency, 

Kj nondimensional bending-moment factor 

I pVUM^ 

L wave length 

L t aerodynamic lift per unit span of wing due to 


L„ aerodynamic lift per unit span of wing due to 

vertical motion of airplane 
m mass per unit span of wing 

Mj net incremental bending moment at wing 

station j 

rbi 2 

cw n (y— y 3 )dy 

J 'bi2 

mw n (y—yj)dy 





generalized mass of nth mode 
incremental number of g acceleration 
see equation (58a) 

load intensity per unit spanwise length 
see equations (18), (24), and (13) 


distance traveled, 

— t , half-chords 


t, T 

T(w) t T(Q) 








wing area 

frequency-response function 

vertical velocity of gust or random disturbance 

maximum vertical velocity of gust 

forward velocity of flight 

total weight ~of airplane 

deflection of elastic axis of wing, positive 

deflection of elastic axis in nth mode, given in 
terms of unit tip deflection 
distance along wing measured from airplane 
center line 







1 -<t> 

$(w), <*(«) 




response coefficient based on a nt «« 

(J Co 

second derivative of z 0 with respect to s 
second derivative of z v with respect to s 
absolute value of center-line deflection of 
‘ fundamental mode in terms of unit tip 

distance interval, half-chords; also, strain 
nondimensional bending-moment parameter, 

apCoMc 0 

reduced-frequency parameter, 

nondimensional relative-densitv parameter, 
apc 0 S 

mass density of air 

standard deviation; also, distance traveled, 

— r, half-chords 

function which denotes growth of lift on an 
airfoil following a sudden change in angle of 
attack (Wagner function) 
power-spectral-density functions 
function which denotes growth of lift on rigid 
wing entering a sharp-edge gust (Kussner 

circular frequency 

natural circular frequency of vibration of /* th 















, 2x w 2 k 

frequencv, — 

Ld V Cq 





spanwise station 

number of distance intervals traveled 

natural modes of vibration 





| | column matrix when used in matrix equations 

[ ] square matrix 

Dots are used to denote derivatives with respect to time 
primes denote derivatives with respect to^or <r;a bar above 
a quantity denotes the time average; and vertical bars about 
a quantity denote the modulus. 





The following analysis treats the problem of determining 
the stresses that develop in an airplane flying through vertical 
gusts on the assumption that the airplane is free to respond 
only in vertical motion and wing bending. The case of the 
transient response to arbitrary gusts is considered first. 
A subsequent section is then devoted to the case of random 
disturbances in which explicit consideration is given the 
continuous nature of atmospheric turbulence. 

Equations of motion. — It is convenient to treat the problem 
simply as one of determining the elastic and translational 
response of a free-free elastic beam subject to arbitrary dy- 
namic forces. For dynamic forces of intensity F per unit 
length, the differential equation for wing bending is, if struc- 
tural damping is neglected, 

d 2 „ r bhv .. . „ 
5 El ^ rj = — mw+F 

z>r V 

( 1 ) 

where w is the deflection of the elastic axis referred to a fixed 
reference plane. The task of determining the deflection that 
results from the applied forces Fmay be handled conveniently 
by expressing the deflection in terms of the natural free-free 
vibrational modes of the wing. 

The wing deflection is thus assumed to be given by the 

'w=d 0 w () -{-a l w l J ra 7 W2 J r ... ( 2 ) 

where the a*’s are functions of time alone, and the w,,’s 
represent the deflections of the various modes along the 
elastic axis of the wing, each being given in terms of a unit 
tip deflection. In equation (2), w 0 represents the rigid-body 
mode and has a constant unit displacement over the span; 
the other w's are elastic-body modes which satisfy the differ- 
ential equation 

(4) and (5), the following basic equation results: 

/ b/2 

Fw n dy (7) 

- 6/2 

which allows for the solution of the coefficients a n if the 
applied forces F are known. This equation applies for the 
translational mode n=0, for which case w 0 =0, as well as for 
all the elastic-body modes. The quantity M n appearing in 
the equation is commonly called the generalized mass of 
the nth mode. 

For the present case of the airplane flying through a gust, 
the force F is composed of two parts: a part designated by 
L v due to the vertical motion of the airplane (including both 
rigid-body and bending displacements) and a part L g resulting 
directly from the gust (this latter part is the gust force which 
would develop on the wing considered rigid and restrained 
against vertical motion). On the basis of a strip type of 
analysis, these two parts are defined as follows: 

F=L,+L,= —^pcvj w [ 1 — <t> (£— T)](ir+^f)cV u\p{t—T)fh 


where f=0 is taken at the beginning of gust penetration. 
!—<£(<) is a function (commonly referred to as the Wagner 
function) which denotes the growth of lift on a w ing following 
a sudden change in angle of attack and for two-dimensional 
incompressible flow is given by the approximation 

v v 

— 0 . 09 — t - 0 . 6 — t 

[1— *«)L,-„ = 1— 0.165e c — 0.335e c (9) 

and i>(t) is a function (commonly referred to as the Kussner 
function) which denotes the growth of lift on a rigid wing 
penetrating a sharp-edge gust and for two-dimensional 
incompressible flow is given by the approximation 


by 2 


, , , — 0 . 26 — f -2 — / 

[yp (01^-co = l — 0.56 c — 0.5* c 00) 

and the orthogonality condition 

X b/2 

mw m w H dy~ 0 (m^n) (4) 

-»/ 2 

=M n (m=n) (5) 

In accordance with the Galerkin procedure for solving 
differential equations, equation (2) is first substituted into 
equation (1) to give, after use is made of equation (3), 

a l «i 2 mW|+a 2 a> 2 2 mtc 2 -f . . . = — m(d 0 tc 0 + . . .) + i r 

( 6 ) 

Xow if this equation is multiplied through by w ny then is 
integrated over the wing span, and use is made of equations 

Figure 1 is a plot of equations (9) and (10). 

An additional term which involves the apparent air mass 
should be included in equation (8) ; this mass term is inertial 
in character and may be included with the structural mass 
(see ref. 16) although it is usually small in comparison. The 
lift-curve slope a may be chosen so as to include approximate 
overall corrections for aspect ratio and compressibility effects. 

The remainder of the analysis is restricted to uniform 
9panwise gusts and the assumption is made that the response 
will be given with sufficient accuracy by considering only 
two degrees of freedom: vertical motion and fundamental 
wing bending. On this basis, if w as given by the first two 
terms in equation (2) is substituted into equation (8) and 
the resulting equation for F is substituted into equation (7). 
the following two response equations result when n is set 



Figure 1. — Unsteady-lift functions (see eqs. (9) and (10)) where, for a sharp-edge gust, the gust force /(*) 

equal lo 0 and 1, respectively: 

a 0 =— (^ 0 +^ [ 1 — (t r)] dr+ ilf (f— r) dr 


2X1\ » , wi 2 2A/i 


( 11 ) 

il'(f “•+! * + 

r) t/r (12) 

where (because of mode symmetry) 


y — 2 I c dy 


*Si — 2J cw x dy ► (13) 


S 2 =2J cw*dy 

Equations (11) and (12) may be put in convenient non- 
dimensional form by introducing the notation 

2V . 

8 — — t 



<r = — r 






where c 0 is the midspan chord of the wing and C is the maxi- 
mum vertical velocity of the gust. With this notation, 
equations (11) and (12) may be written 



r#" = — 2 f (V , +'Y‘ i ")ll— "M* - p^(x— <7 

)// (7 

Mi — 2 J (/*i (1 ~ 0( v — <r ) | r/cr 4- 

wlie re 

x — a) da M T ! 

_S47„ -t 

_ SM t 

M ‘ ftpc.^S 


'* 1 = 






and a prime denotes a derivative with respect to a. Equa- 
tions (10) and (IT) are the basic response equations in the 
present analysis. The live parameters appearing in these 
equations and given by equations MS) depend upon the 
forward velocity, air density, lift-curve slope, and the air- 



plane physical characteristics: the wing plan form, wing 
bending stiffness, and wing mass distribution. Experience 
has shown that variations in the physical characteristics 
cause significant variations in the first three of the five 
parameters, while the last two vary only to a minor extent. 
The first three are therefore the most basic parameters; mo is 
a relative-density factor, frequently referred to as a mass 
parameter, and is associated with vertical free-body motion 
of the airplane; p,, similar to ji<>. is the mass parameter 
associated with the fundamental mode; and X, by its nature, 
may be interpreted as a reduced-frequency parameter similar 
to that used in flutter analysis. 

It is significant to note that, if any one of the three quan- 
tities : 0 , and u appearing in equations (16) and (17) is 
specified or known, the other two may be determined. Thus, 
if the gust is known, the response may be determined or. 
conversely, if either r 0 or z t is known; the gust may be 
determined. A useful equation relating r 0 and ~ x may be 
found by combining equations (16) and (17) so as to eliminate 
the integral dealing with the gust. The result is the equation 

J t ( - X-' r, ) + 2 (^- /-, ) J^’ r, ' ' ( l - 0 (* -,)]</*=*,-„" (19) 

which is used subsequently. 

It may also be of interest to note that MoV', in effect, 
defines a frequently used acceleration ratio. From equa- 
tions (15) and (14), the rigid-body component of the vertical 
acceleration may be written 




or, when expressed in terms of the incremental number of </ s, 

if Cog 

An acceleration factor An, based on quasi-steady flow and 
peak gust velocity is now introduced according to the 

'=\ PSV y 


The ratio is thus found to be 




— M<KO 

Where the gust shape is represented analytically and the 
imsteadv-lift functions arc taken in the form given by equa- 
tions (0) and (10), solution of the response equations may he 
made by the Laplace transform method, hut such a solution 
is more laborious than desired. Therefore, a numerical 
procedure which permits a rather rapid solution of the 
equations has been devised for the rase of discrete-gust 
encounter and is presented in a subsequent section. It may 
be well to mention, however, that the response equations 
are suitable for solution by some of the analog computing 

Bending stresses. — The bending moment and, lienee, the 
bending stresses that develop in the wing due to the gust 
may he found as follows: The right-hand side of equation 

(1) defines the loading on the wing; suppose that this loadin'; 
is denoted by p % then 

p— —mw-hF 

By use of equations (2) and (8) and the notation of equa- 
tions (14) and (15), this equation becomes 

p= — W 4 {Zo”+z l "w,)— apcVC ( ('„"+ri")4>i) 

Co Jl) 

[l — <f>(ft — pci J u'}f/(x — <t)'I<t 

where, as before, only the first two deflection terms have 
been retained. If the moment of this loading is taken about 
a given wing station, sav y h the following eq nation for 
incremental bending moment at that station will result* 

/• ft /2 

-'/j= p(y—y,)'lu 


= - 4W (M mi z 0 ”+M„^")-apVr 

Co 0 1 Jo ’ 

d/ fl 2i")[l — <*>(# — u'4s(x—o)(fo (20; 

where the A/’s bearing double subscripts are first moments 
defined as follows: 

/•A/2 /'A/2 *> 

Mm 0 = V/ e = c{y—y } )<hi 

Juj Jyj 

/•A/2 /• A/ 2 

A/ W( = mw,(y— yjfly M r = cw x (y—y t )tly 

Jtlj Jvj 

r (2n 

and )jj is the station being considered. Dividing equation 

(20) by the quantity ® pVfWf^ gives the following equation 

which is considered to define n bonding-moment factor K, at 
wing station y } 

K— * 

\ ) “J. \ " + U/‘ ) 

1 1 — 0 (.<— <r)|f/ff+ J /'(x — a)tla 

( 22 ) 

The factor may be regarded as the maximum 

aerodynamic heading moment that would he developed 
by the gust under conditions of quasi-steady flow and with 
the wing considered rigid and restrained against vertical 
motion at the root. The bonding-moment factor K } may 
thus he seen to he the ratio of the actual dynamic bending 
moment that occurs to this quasi-steadv bending moment 
aiul therefore mav he regarded as a response or an alleviation 

A more convenient form for the bending-moment factor 
may be obtained by solvin'; equations (16) ami <17> 

simultaneously for the quantities I — <&(*— cn\do and 



J* zfll — *( 5 — <r)]d<r and substituting these values into 

equation (22) . With these operations the following equation 




| pVUMc 0 

=dz 0 "+ez l "+k\ 2 z l 


, r 3 ri — r 2 
a 9 Mo Vo 

rf— i* 

_ r i ~~ 

n 2 — r 2 

Ml -1 ?! 


The derivation proceeds on the basis that the response 
due to a given gust is to be determined. The airplane, just 
before gust penetration, is considered to be in level flight 
and, hence, has the initial conditions that the vertical dis- 
placement and vertical velocity are both zero. These con- 
ditions mean that z 0 , z Jf z 0 ', and Z\ are all zero at 5 = 0. 
The gust force can be shown to start from zero and, therefore, 
the additional initial conditions can be established that z o' 
and 2," are also zero at 5 = 0. By the numerical procedure, 
solution for the response at successive values of « of incre- 
ment « will be made and, for the case being considered, it is 
found advantageous to solve directly for the accelerations 
rather than the displacements. 

In order to make the presentation more compact, the 
following notation is introduced: 



_ A — r 3 
ri 2 — r 2 r 



'M 0 




8 M mi 

m ~~apcoMc 0 j 

It is seen that, when bending moments are being determined, 
three additional basic parameters (eqs. (24)) appear. The 
similarity of rjo and to no and mi is to be noted; first moments 
of masses and areas are involved rather than masses and 

Reduction to rigid case. — It may be of interest to show the 
reduction of the response equation to the case of the air- 
plane considered as a rigid body. Thus, if z x is equated to 
zero in equation (16), the following equation for rigid-body 
response is obtained: 

^, 20 ," = — ?J‘ V[l — <*>(«— <r)]d<r+ jj^{s—<r)da (25) 

If 2 " is set equal to zero in equation (22) and use is made 
of equation (25), the following equation for the bending- 
moment parameter for the rigid-body case is obtained 

^ r =(Mo-%)V (26) 

where 2 0 / is the nondimensional acceleration of the airplane 
considered as a rigid body. 


The case of discrete-gust encounter. — In this section a 
rather simple numerical solution of the response equations 
(16) and (17) is presented for the case where discrete gusts 
are suddenly encountered. The procedure is readily adapted 
to either manual or punch-card -machine calculations. 



/(5)=J jjyf/(8 — (r)d(j (27b) 

With this notation, equation (16) would appear simply as 

Mo**— 2 /: (a+ r v fi) 0(5— <r)d(r 4-/(5) (28) 

In accordance with numerical-evaluation procedures, the 
interval between 0 and 5 is divided into m equal stations of 
interval «so that5=m«. The product of (a+ri/9) and 9(s — <r) 
is assumed formed at each station and, with the use of the 
trapezoidal method for determining areas, the unsteady-lift 
integral in equation (28) may be written in terms of values 
of a and ft at successive stations as follows, where the mth 
station corresponds to the value s: 

J (a4*ri/3)0(5— ff)d<r=e ^0 m _iai + 0 w _ 2 a 2 + . . . + 0ia m _i + 

^ 0 o a m ^ + €r 1 ^0 m _i0i + 0 m _202 + • * • -|-0i0«-i+2 

in which 0 O , 0 t , . . . are, respectively, the values of the 
1—* function at 5=0, 5 = e, . . . (ao and j3 0 do not appear 
because of the initial conditions). With this equation, 
equation (28) may be written at various values of s or at 
successive values of m; the result, for example, for w = l is 

Mo<*i — — «0o«i — «/*|0o0i +/i 

and for m= 2, 

Mo<* 2 = — e(20 l CTi+0o«2) — er 1 (20101 +0002) f 2 

where /, and / 2 are the values of the gust-force integral at 
s=€ and s=2*. 


The equations thus formed may be combined in the following matrix equation: 
Mo + 0o« ot\ p r,0 O € 

2ft* Mo + ft* a 2 2 nfte r x 6& 

2ft* 2 fte Mo+ft* «3 2rjft< 2 /^fte r.i 







L 2 ft*-i* 2 ft_ 2 * . . . Mo+^Jl^wl I2rxe m - X < 2 r l $ m _ 2 € . . . J I ft I 1 /« j 

which may be abbreviated 

H]|«| + [B]|P| = |/| (30b) 

The simplicity of the matrices A and B , and all square matrices to follow, is to be noted; the matrices are triangular and 
all elements in one column are merely the elements in the previous column moved down one row. Thus, only the elements 
in the first columns have to be known to define completely the matrices. 

Xow instead of considering directly the second response equation, equation (17), it is expedient to consider equation 
(19). According to the derivation presented in appendix A, the value of z x at $=rru may be approximated in terms of the 
past-history value of Z\ n by the following equation: 

^ pi_ Da 

+ • • • +2ft_ 2 +ft 

m — 1 “{“ g 

where ft, ft, are the values of z v " at $=*, s = 2c, . . . . If this equation is used to replace z x in equation 

(19) and the unsteady-lift integral is manipulated similarly to the integral in equation (28), equations are obtained for suc- 
cessive values of m which involve only the unknowns a and ft The results may be combined in the following matrix equation: 

?( 1 + vMS _r, ) < ' 04 
? xV+2 G? -ri ) 9it 

2 — XV-f-2 
r i 



,7 kV+2 

j^(m~l)^X 2 « 2 +2^~ri^ wl - 1 e (m— 2)^XV+2^“r^ft_ 2 € 
which may be written 




oc 2 


— Mo 






[C] \0\ = Ho\a\ 

The square matrix [C] is seen to be similar to the other square matrices in that it is triangular with all the elements in 
one column made up of the elements in the previous column moved down one row. 

An equation in |ft alone is obtained by substituting \a\ from this equation into equation (30) to yield 

{i[yl][C] + [B]j|/3| = [Z)]|/9| = |/| ( 33 ) 

which is the basic response equation relating 0 (that is, 2 ,") to the gust force. This equation represents a system of 



linear simultaneous equations where the order of the matrix 
is arbitrary; that is, the equations may be written up to any 
desired value of s = The solution for response can there- 
fore be carried on as far as desired. Fortunately, the equa- 
tions are of such a nature that simultaneous solution is not 
required. As mentioned, each of the matrices [A], [B\ t and 
[C] is triangular with all elements 0 above the main diagonal 
and with all elements on the main diagonal of each matrix 
equal; therefore, the main diagonal elements of [D] will also 
all have the same value and the elements above this diagonal 
will be 0. If each element on the main diagonal of [D] is 
denoted by d x and [Di] is the matrix [D] with the main diagonal 
elements replaced by 0’s, then 


With this equation, equation (33) may be written 


Expanded, this equation has the form 







d t 0 



f 3 

d% di 0 




/ 4 


di d^ di 0 


d X 



f 5 

di d 3 di 0 




It can be seen that a step-by-step solution for the successive 
values of 0 may now be made; that is, ft is solved for first, 
then, with ft established, ft is solved for, and so on, as far as 
is desired. With the value of \0\ thus established, solution 
for \a\ may now be made directly from equations (32). 
Values of the displacements z 0 and z i may be obtained directly 
from a and ft z x may be obtained from equation (31); and 
z 0 may be obtained from this same equation with 0 replaced 
by a. 

Some mention should be made with regard to the selection 
of the interval «. A rough guide to use in selecting € can be 
obtained by considering X, which appears as the characteristic 
frequency in most response calculations. The period based 

on this frequency would be T 9 =~ Experience has shown 

that an interval in the neighborhood of 1/12 of this period 
yields very good results (in general less than 1 percent error) ; 
accordingly, a reasonable guide in choosing e would be the 

equation Some convenient value near that given by 

this equation should be satisfactory; in general, it will be 
found that c may be 1 or greater. 

The procedure thus outlined provides a rather rapid eval- 
uation of the response due to a prescribed gust. With the 
response thus evaluated, the bending moment at any value 
of .s or the complete time history of bending moment may be 
found by application of equations (23). 

As a convenience in making calculations, a summan' of 
the procedure developed in this section has been made and 
is given in appendix B. Curves of the value of the gust 
force, equation (27b), are also given for three different types 
of discrete gusts: sine gusts, sine 2 gusts, and triangular 

As a final word, it should be evident that, if response 
values for either z 0 " or z x " are known, the gust causing this 
response can be found bv suitable manipulation of equations 
(30) and (32). Thus, if z 0 " is known, 0 in equations (30b) 
and (32b) may be eliminated to give the equation 


Direct substitution of z 0 " in this equation allows |/j to be 
determined. In most practical cases the second term in 
equation (30b) contributes only a small amount and may be 
dropped with little resulting error in the gust force. The 
equation for |/| is then simply 


The case of continuous-sin usoidal-gust encounter. — Of 

primary importance in making continuous-turbulence studies 
is the response of the aircraft to a continuous sinusoidal gust. 
A reduction of the response equations to this case is therefore 
now made. 

Where the gust is sinusoidal with frequency w, the quan- 
tities u, z 0t and z x may all be taken proportional to e ik \ where 

and it may be shown that equations (16) and (19) 

reduce to (eq. (19) is chosen in place of eq. (17) purely for 

mo *>" = -2 W+wMF+iGi+p ( P+iQ ) (36) 

- U,"+X 2 s,)+2 z,'(F+iG)=^z n " 

/*] \/‘l / 


where F(k) and G(k) are the in-phase and out-of-phase oscil- 
latory lift coefficients used in flutter work and P{k) and ()(k) 
are the similar in-phase and out-of-phase lift components on 
a rigid wing subjected to a sinusoidal gust (see, for example, 
ref. 17). 

Xow let the gust velocity and the motion be represented 
by the real parts of 

u=Ue lk * \ 

Zq — Zo« 
Z L 

0 / 


where Z 0 and 7 X may be complex. With these equations 
equations (36) and (37) become 

F ?) Zo+F ( - '' 1 T+ in t) Zl =P+ iQ 






These equations yield 

7 (R*~\~iS t )(P -H@) 




£ 2 (A, + iA 2 ) 




2 G 

R\ — Mo — 

P 2(t 

s ‘= s '-+(3P-s) s 

\ 2<? 

V * 


5i= t 


Si ~ Tl T 

\ 2F 



Ai — — 5 i 1 S 4 — i?2-^3 

Aj— ^?iiS , 4“f"/?4iS , l — R$S2 

in which Z? 5 has been included because it appears later. 
With Z 0 and Zj established, the various response quantities 
of interest may be determined. Those used frequently are 
(1) the rigid-body component of acceleration z 0 ", (2) the 
acceleration at the fuselage center line 

s"(0) = V'+2i , 'wi(0) = e 0 " -te/' 

where 6 is the absolute value of the fundamental-mode 
deflection at the fuselage in terms of a unit tip amplitude, and 
(3) the bending-moment factor K j =dz i) " +eZi" +h\ 2 z u see 
equation (23a). In accordance with equations (38), these 
quantities may be written as the real parts of 


2 "(0)=-/P(Z o -5Z 1 )t < *’ 
K,=—kr ^/Z 0 +(e-A p) Z,J , 


With the use of equations (41) and (42), these equations 

(R<+iS 4 ) (P+iQ ) iAt , 

Ai-Ra 2 

m ,ti. _ (R «+ &R j+ ( P 

‘ ' ■ ~ A,-HA 2 ' e 

w <KR,+iS<)(P+iQ) it, 

1 A 1 + ,-A,' * 


The squares of the amplitudes follow directly from these 
equations and are listed below since they play a primary role 
in many applications 

■ 2 _(R, 2 +s 4 W 2 +Q 2 ) 

|2 ° 1 aP4^7 


_*m>i*-[( /? ‘+ a/? 3 ) 2 +‘W 2 +Q 2 ) 

2 ^ a,’+a 2 2 

P(R s '+S t I )(I”+Q *) 
{K>1 aTTa? 

It is worthwhile at this point to mention that a good approx- 
imation exists for the quantity P 2 -\-Q 2 which appears in all 
three equations. This quantity reflects the force on the 
airplane due directly to the sinusoidal gust and for two- 
dimensional incompressible flow is approximated with good 
accuracy by the expression (see ref. 8) 

pi+ ^i +h 

Two other quantities which are used frequently in appli- 
cations are now presented. These two quantities are the 
acceleration and bending-moment factor that apply when 
the airplane is considered as a rigid body, that is, when it is 
considered to have only the degree of freedom of vertical 
motion. The equation for rigid-body response can be 
obtained directly from equation (39) by setting Z t — 0. 
With the aid of the resulting equation it may be shown that 
the square of the amplitude of the rigid-body acceleration is 


P 2 +Q 2 



Through use of equation (26), the rigid-body bending- 
moment factor may be written 

|^ r r = Cuo-^) 2 |Zo/| 2 (51) 

As a closing remark to this section, it may be said that the 
computation of the response to a continuous sinusoidal gust 
is actually quite an easy task, the amount of work involved 
being very small in comparison with that involved in a 
discrete-gust calculation. All that is necessary is to evaluate 
the response quantity of interest, equations (46) to (48), 
through means of the coefficients given by equations (43), 
with k taken equal to the reduced frequency of the sinusoidal 
gust under consideration. Because the computation is so 
straightforward, no summary is given as in appendix B for 
the case of discrete-gust encounter. 


In order to provide an illustration and give an idea of the 
accuracy of the present analysis, the response to a sharp- 
edge gust of the two-engine-airplane example considered in 
reference 16 was determined. The weight distribution over 
the semispan, the wing-chord distribution, and the funda- 
mental bending mode are shown in figures 2, 3, and 4. The 
frequency and deflection of the fundamental mode were 
calculated by the method given in reference 18. The solu- 
tion is made for a forward velocity of 210 mph and a gust 
velocity of 10 ft/sec. 



The lift-curve slope used in reference 16 was 5.41; to be 
consistent, the same value was used herein. Furthermore, 
the unsteady-lift function used for a change in angle of attack 
in the example presented in reference 16 was given by the 

(1“<£)a-«— l*”0.36l€ -0 ' 381 * 

rather than by equation (9). Thus, this equation was also 
used herein. The gust unsteady-lift function used was that 
given by equation (10). 

The various physical constants and the basic response and 
bending-moment parameters are given in table 1 ; the values 

Figure 2.— Semispan weight distribution for the two-engine airplane 
of example. 

of the unsteady-lift function and the values of the gust force 
are listed in table 2. The matrices [/l], [B], and [C] used 
in the solution are given in table 3. 

The solution for response is shown in figure 5 (a) where the 
deflection coefficients a 0 and a x in inches are plotted against 
distance traveled in half-chords. The corresponding deflec- 
tion quantities for the example given in reference 16 were 
determined and, for comparison, are also shown in the figure. 
A similar comparison is made in figure 5 (b) for bending 
stresses at the fuselage and engine stations, stations 0 and 1 
from reference 16. The agreement is seen to be good. 


w ; lb 

5, sqft - 

6, in. - 

c*, in. 

p, slugs/cu ft 

V, ft/sec 

V, ft/sec 

fsec - - 



|M— - - 


/fuselage station — - 

r, iengine station..- - 

j 1 fuselage station 

^engine station - - 

/fuselage station 

1,1 \ engine station 

•* , _Jfuselage station 

I ,wl * i.englne station 

*z here denotes distance from neutral axis to extreme fiber. 

37. 430 

0 . 0023 * 



0. 020* 
5. 41 
54. 15 
0 9043 
0. 4353 
0. 2181 
0. 1358 
0. 347 
23. 49 
10. 19 
3 555 
3. 391 
0. 00659 



9m Or (l-*)4-4 

for + 

. 0 




. 7534 

. 377 



. .547 




i 4 

. 9214 


i * 

. 9463 



. 9633 




. 798 1 






. 845 1 


.1 Matrix 

r64. 799 

1.5068 64.799 
1.6630 1.5068 

1. 7698 1. 6630 

1. 8428 1. 7698 

1.8926 1.8428 

1.9266 1 8926 

L. 9498 1. 9266 

1.9658 1.9498 

_ 1.9766 1.9658 

r 0. 1394 

. 3286 0. 1394 

.3627 . 3286 

.3860 . 3627 

.4019 .3860 

.4128 .4019 

.4202 .4128 

. 4252 . 4202 

. 4287 . 4252 

_ .4311 .4287 

r 4.5367 

1.3954 4.5307 

2.2445 1.3954 

3. 0735 2. 2445 

3. *889 3.0735 

4. 0949 3. *8*9 

5. 4947 4. 0949 

6. 2900 5. 4947 

7. 0824 6. 2900 

_ 7.8726 7.0*24 

64. 799 

1.5068 64.799 
1.6630 1-5068 

1.7698 1.6630 

1.8428 1.7698 

1.8926 1.8428 

1.9266 1.8926 

1. 9498 1. 9266 

0. 1394 

. 3286 0. 1394 

. 3627 . 3286 

.3860 . 3627 

.4019 . 3860 

.4128 .4019 

.4202 .4128 

. 4252 . 4202 

4. 5367 

1.3954 4. 53*17 

2.2445 1.3964 

3.0735 2.2445 

3. **8« 3. 0735 

4. 6949 3. 8889 

5.4947 4.6949 

6. 2900 5. 4947 


1. 5068 64. 799 
1.6630 1. .5068 

1.7698 1.6630 

1.8428 1.769* 

1.8926 1.8428 

li Matrix 

0. 1394 

.3286 0.1394 

. 3627 . 3286 

. 3*00 . :W27 

.4019 .3*00 

.4128 . 4019 

C Matrix 

4. .5367 

1.3954 4.5367 

2. 2445 1 . 3954 

3. 0735 2. 2445 

3. *889 3. 0735 

4. 6949 3. *8*9 

64. 799 

1.5068 64.799 
1.6630 1.5068 

1.7698 1.6630 

0. 1394 

.3286 0.1394 

. 3027 . 3286 

. 3860 . 3627 

4. .5307 

1. 3954 4. 5307 

2.2445 1.3954 

3.0735 2,244.5 

64. 799 

0. 1394 

4. 5307 
I 3954 

64. 79 9 J 

0. 1394 J 

4. 5307 J 




Although the unsteady-lift functions for two-dimensional 
unsteady flow are presented, the method is general enough 
so that the unsteady-lift functions for finite aspect ratio, for 
subsonic compressible flow, and for supersonic flow may be 
used as well. (See refs. 3 and 17 to 22.) 

Since the numerical method for the case of discrete-gust 
encounter is based on an integration procedure, it possesses 
the desirable feature that a fairly large time interval may 
be used and good accuracy still be obtained. As an accuracy 
test, solutions of equations (16) and (17) were made for 
several cases by the exact Laplace transform method as 
well as by the numerical process, in which process the time 
interval was selected according to the rule of thumb sug- 
gested. When the results were plotted to three figures, 
the difference between the two solutions was barely 

Additional bending modes could be included in the 
analysis but this refinement is really not warranted. Some 
calculations made with additional modes gave results which 
differed only slightly from the results obtained when only 
the fundamental mode was used. The good agreement of 
results obtained for the example with the results obtained 
by the more precise method given in reference 16 also 
illustrates this point. Furthermore, if additional degrees 
of freedom are to be used, it would appear more important 
to extend the analysis to include wing torsion and airplane 
pitch and, also, to include the case of nonuniform spanwise 
gusts. Torsion undoubtedly becomes important for speeds 
near the flutter speed, and pitch would appear important 
for cases where low damping in pitch is present. This latter 
point has been borne out by some investigations which show 
that there is a marked increase in gust loads as the damping 
in pitch is decreased. However, it is the intent of this 
analysis to treat the effects of wing bending flexibility and 
it should be sufficiently satisfactory for speeds at least up 
to the cruising speeds and for airplanes having good longi- 
tudinal damping characteristics. 


The approach given in the previous section works well 
for gusts which are either isolated or which are of a con- 
tinuous-sinusoidal type. It also works for gusts which are 
of a random-continuous nature, such as exist in the atmos- 
phere. For this case, however, the approach is not very 
practical, first because it is questionable whether an appro- 
priate or representative time history of atmospheric gust 
sequence could be established, and second because for any 
long gust sequence the amount of computational work 
involved is prohibitively large. It is therefore desirable to 
turn to other means for treating realistic turbulence condi- 
tions, with the view of having a technique that has general 
applicability and is mathematically tractable. 

One such procedure which suggests itself for treating the 
case of random continuous turbulence and which is at present 
receiving much attention makes use of the concepts and 
techniques of generalized harmonic analysis (see, for example, 
refs. 6 to 10). These methods permit the description of the 
random-atmospheric-turbulenre disturbance and the associ- 
ated airplane response in analytic form by means of the 

so-called “power-spectral-density function/’ A brief review 
of the technique is considered pertinent. If u(t) represents 
a random disturbance or a system response quantity to this 
disturbance (such as the atmospheric vertical velocity and 
resulting structural response considered herein), then the 
power-spectral-density function <t>(«) is defined as 

<I>(a>) = lim I f u(t)e~ iul dt\ (52) 

t— ® l \ J -T 1 

where w is frequency in radians per second, and the bars 

J r 

which is known as the Fourier transform of u(t). An 
equivalent and more useful expression for <t>(w) can he 
derived and is 

<t>(<j)= : - I R(r) COS air dr (53) 

r Jo 

where R(t) is the autocorrelation function defined by 

R(t)= lim —j, J u(t)u(t+r)dt (54) 

A useful property of <t>(aj) is that 

do>=Mean square = u 2 {t) — R{0) 

(55 ) 

The quantity u 2 (t ), or <r 2 , the time mean square, provides 
a measure of the disturbance energy per unit time and has 
thus been characteristically termed the power, as a carryover 
from its early application in the fields of communications 
and turbulence, where it often had the dimensions of power. 
Thus, 4>(w) has, in turn, been termed the energy or 
power spectrum. In this form, the element <f>(ud d u> gives 
the- contribution to the mean square of harmonic components 
of u(t) having frequencies between « and w-Mu. 

Now a particularly useful and simple relation exists for 
linear systems between the spectrum of a disturbance and 
the spectrum of the system response to the disturbance (sec 
refs. 8 and 23). This relation is 


3> 0 (oj) =4> £ (w) T' 2 (u ) ) 


$ tf (w) output spectrum 

<I> £ (w) input spectrum 

T(oj) amplitude of admittance frequency-response func- 

tion which is defined as the system response to 
sinusoidal disturbances of various frequencies 

It is precisely because of this equation that the response 
to a continuous sinusoidal gust was derived in the previous 
section. The equation indicates that the response at a 

given frequency depends only on the input and the system 
admittance at that frequency, which is plausible for linear 

A significant point to note here is that, despite the fact that 
continuous random disturbances are under consideration, the 
response equation (56) turns out to be surprisingly simple 
and easy to apply. This fortunate outcome is undoubtedly 
one of the eonsequenees of working in the frequency plane 
rather than the time plane. Nevertheless, even though the 



frequency plane is involved, it is still possible in particular 
cases to determine a number of statistical characteristics of 
the disturbance or response time histories which are of 
interest. For example, the root-mean-square value <r, which 
may be obtained directly from the spectrum in accordance 
with equation (55), provides a useful linear measure of the 
disturbance or response intensity. Further, in the particular 
case in which the function u(t) has a normal or Gaussian 
probability distribution with zero mean, the probability 
density is given by 

p(y)=— (57) 

<r\ 2 t 

Also, S. 0. Rice, in reference 24, has derived for the case in 
which the disturbance function is completely Gaussian a 
number of relations which appear useful in aeronautical 
applications and which are particularly significant for fatigue 
studies. One of the more important expressions is for the 
average number of peak values (maximums) per second that 
are above a given value of u. For the larger values of u 
(say u>2<r), the expression is 



There is some indication, as described in reference 6, that 
airplane gust loads may tend to have a normal distribution. 
Hence, use is made of these equations subsequently in the 
application to the flexibility studies. 

As a schematic illustration of the application of equation 
(56) to the problem of airplane response to gusts, figure 6 lias 
been prepared. The top sketch in this figure is the input 
spectrum and, in this case, represents the spectrum of at- 
mospheric vertical velocity. The frequency argument ft, 
which is 2 it divided by the wave length L, is introduced in 
place of a) because gust disturbances are essentially space 

INPUT: chorocterizes the 

A t > t meon-squore value of 
gust velocity 

<Tl = r/: 

characterizes the 

disturbances rather than disturbances in time. The second 
sketch T 2 (ft) represents the amplitude squared of a specified 
airplane response, such as the airplane normal acceleration 
(eq. (46)) to sinusoidal gusts of unit amplitude and of fre- 
quency ft. ^Xote that «, k, and ft are related as follows: 

0 , = ^!)= . ] Tliis function introduces the characteristics 

Co / 

of the airplane, the various modes usually showing up as 
peaks such as the free-body and fundamental wing-bending 
modes illustrated. The output spectrum $ 0 (Q) is obtained 
in accordance with equation (56) (this equation applies 
whether the argument is to or ft) as the product of the first 
two curves and gives, for example, the spectrum of normal 
acceleration or the spectrum of stress, depending upon what 
quantity is chosen for the frequency-response function. This 
output spectrum indicates the extent to which various fre- 
quency components are present in the response, and, further, 
it allows for the determination of various statistical proper- 
ties of the response time history, such as are given by equa- 
tions (55), (57), and (58). 


A number of flight and analytical studies have been made 
which deal with the effect of wing flexibility on the structural 
response of an airplane in flight through rough air (see refs. 
10 to 12 and 25 to 28). The primary results of these studies 
are summarized in this section. Specifically, the following 
material is covered. The significant results of flight tests 
are given. Studies made on the basis of single- or discrete- 
gust encounter are then reviewed and the extent of the 
correlation with flight-test results is indicated. Finally, 
some analytical work on the more realistic condition of 
continuous-turbulence encounter is presented and corre- 
lation with flight tests shown. 


From an analytical point of view, several measures may 
be devised to indicate the extent to which flexibility effects 
are present in any airplane. Generally these measures 
indicate how a particular structural-response quantity (such 
as acceleration) for the flexible airplane compares with what 
this response would be if the aircraft behaved as a rigid body, 
a comparison of z 0 " with 2 0r ", for example. For the correla- 
tion purposes of the present report, however, the flexibility 
measures have been confined largely to the two types used 
in flight tests. One of these measures involves a comparison 
of the peak incremental accelerations developed at the fuse- 
lage with the peak incremental accelerations at the nodal 
points of the fundamental mode (see fig. 7), the latter accel- 
eration being considered a close approximation to what the 
acceleration would be if the airplane were rigid. These two 
accelerations are of particular interest because both have 
been considered in the deductions of gust intensities from 

Fic.ure 7. — Fuselage and nodal accelerations. 



measured accelerations; they are different, in general, as are 
all accelerations along the wing, because of structural flexi- 
bility, particularly wing bending. The other flexibility 
measure involves a comparison of the actual incremental 
wing stresses with what these stresses would be if the 
airplane were rigid. Since it is, of course, not possible to 
obtain the rigid-body reference strains in flight, some near- 
equivalent strain must be used. The general practice has been 
to assume that the rigid-body strains are equal to the strains 
that would develop during pull-ups having accelerations 
equal to the accelerations that are measured at the nodal 
points during the rough-air flights, and this practice has been 
followed herein. 


In order to establish what the numerical values of these 
flexibility measures are in practical cases, flight tests were 
made in clear rough air with the three airplanes shown in 
figure 8 and designated A, B, and C as shown. References 
25 to 28 report some of these flight tests. These airplanes 
were chosen because they were available and because the\ 
were judged to be fairly representative of rather stiff, 
moderately flexible, and rather flexible airplanes, respectively. 
In this flexibility comparison, the factors which are considered 
to signify an increase in flexibility effects are higher operating 
speeds, lower natural frequencies, and greater mass in the 
outboard wing sections. Figure 9 shows the type of accelera- 


(a) Airplane A. 

Figure 8.— Three-view sketches of test airplanes. 

(b) Airplane H. 

Figure 9. — Acceleration measured in clear rough air. 

(c) Airplane <\ 


tion results obtained from these flights. The ordinate refers 
to peak incremental acceleration at the fuselage and the 
abscissa refers to the peak incremental acceleration at the 
nodal points. Although only positive accelerations are shown 
in this illustration, a similar picture was obtained for nega- 
tive acceleration values. The solid line indicates a 1 to 1 
correspondence; whereas the dashed line is a mean line 
through the flight points. The slope of this line is the ampli- 
fication which results from flexibility; thus, the fuselage 
accelerations are 5 percent greater on the average than the 
nodal accelerations for airplane A, 20 percent greater for 
airplane B, and 28 percent greater for airplane C. It is to 
be remarked that the picture is not changed much if given 
in terms of strains; that is, if the incremental root strains 
for the flexible case are plotted against the strains that would 
be obtained if the airplane were rigid, similar amplification 
factors are found. 


In an attempt to see whether these amplification factors 
could be predicted by discrete-gust studies, some calculations 
were made by considering the airplane to fly through single 
sine gusts of various lengths. The calculations were made 
by the discrete-gust analysis presented previously. The con- 
ditions used for speed, load distribution (payload and fuel), 
and total weight were similar to those used in the flight tests. 
Some of the significant results obtained are shown in figure 
10 (see ref. 12 for additional related results). The ordinate 
is the ratio of the incremental root strain for the flexible 
airplane to the incremental root strain that is obtained for 
the airplane considered rigid. The abscissa is the gust-gradi- 
ent distance in chords, as shown in the sketch. The curves 
indicate a significant increase in the amplification or response 
ratio in going from airplane A to B to C, It may be 
remarked that the amount of amplification is, in fact, 
related to the aerodynamic damping associated with wing- 
bending oscillations. This damping depends largely on the 
mass distribution of the airplane and is lower for higher out- 
board mass loadings. The curves thus reflect the succes- 
sively higher outboard mass loadings of airplane B and 
airplane C. 

FifJURK 10. — Strain amplification for single-gnat, encounter. 

The important point to note about this figure is that the 
general level of each curve is in good qualitative agreement 
with the amplification values found in flight. Thus, the 
1.05 value for airplane A roughly represents the average of 
the lower curve, the 1.20 value for airplane B the average 
of the middle curve, and the 1.28 value for airplane C the 
average of the upper curve. A more direct quantitative 
comparison would be available if a weighted average of the 
calculated curves could be derived by taking into account 
the manner in which the gust-gradient distances are dis- 
tributed in the atmosphere. No sound method is available 
for doing this, however, and this overall qualitative compar- 
ison will therefore have to suffice. 

Figure 11 shows what is obtained when calculation and 
flight results are correlated in more detail. In this figure, 
the strain ratio is plotted against the interval of time for 
nodal acceleration to go from the 1 g level to a peak value. 
This interval, when expressed in chord lengths, is slightly 
different from the gust-gradient distance. The flight values 
shown were obtained by selecting from the continuous 
acceleration records a number of the more predominant 
humps that resembled half sine waves and then treating 
these humps as though they had been caused by isolated 
gusts. The agreement seen between the calculated results 
and the flight results is actually surprisingly good when the 
complexity of the problem and the fact that the calculations 
are for a highly simplified version of the actual situation are 
considered. In contrast to the well-behaved single gusts 
assumed in the calculations, the gusts encountered in flight 
are not isolated but are repeated and are highly irregular in 
shape. These factors may well account for the higher ampli- 
fications found in flight, especially in the range of higher 
values of time to peak acceleration; in this range it is to be 
expected that the amplification effects associated with the 
higher frequency components of the irregular gust shapes 
are superposed on the amplification effects of the predomi- 
nant gust length to lead to the higher effective values 



From the results thus far presented, it may be concluded 
that a reasonably fair picture of flexibility effects may be 
obtained with the discrete-gust approach. It is found to 
give good overall qualitative agreement with flight-test 
results and can be used to determine how one airplane com- 
pares with another in respect to the relative extent to which 
these effects are present. Detailed quantitative correlation 
is not feasible, however, since the degree of resolution per- 
mitted by the approach is limited. This is. of course, to be 
expected in view of the limited and unrealistic description 
of turbulence used. 


The procedure given in the section entitled “Treatment of 
Random Continuous Turbulence” was applied in order to 
see what it would yield in the way of flexibility effects for 
the three airplanes used in the rough-air flight tests. The 
spectrum chosen for atmospheric vertical velocity was that 
given in reference 6. Bending stress at a station near the 
root of the wing was chosen as the response variable, and 
evaluation was made for flight conditions representative of 
those used in the flight tests. These conditions are indicated 
in table 4, together with the physical constants and basic 
parameters that apply. (It is remarked that the use of the 
theoretical lift-curve-slope value of 2* in place of more rep- 
resentative values has no serious consequence herein since 
the final results to be presented are in a ratio form which is 
relatively insensitive to the lift-curve slope used.) Figure 12 
shows the transfer functions that were obtained by means of 
equations (48) and (51); for this evaluation the flutter co- 
efficients for two-dimensional incompressible flow and an 
amplitude of the sinusoidal input gust of 1 ft/sec were used. 
The solid curve is for the flexible airplane and the dashed 
curve, for the airplane considered rigid. These curves show 

quite clearly the different bias that each airplane has toward 
various frequency components of the atmosphere. The first 
hump is associated with vertical translation of the airplane 
and the second hump, with wing bending. The spectra for 
bending-stress response, obtained by multiplying the 
frequency-response curves by the input spectrum, of course 
show that the curve for the flexible case overshoots the curve 
for the rigid case by an amount consistent with the frequency- 
response curves. This overshoot is a reflection in the 
frequency plane of the characteristics of the traiisient- 
response curves shown in figure 10. The area of the over- 
shoot is a direct measure of the amplification in mean-square 
bending stress that results from wing flexibility. 



Airplane A 

Airplane B 

Airplane C 

Fuselage loud 

Crew onlv 

> 2 full 

Crow only 

Fuel load 

4 fuii 

1 2 ftlll 


2 MAX) 

33. 470 

105. yon 

i \ mph 

N. sq ft . . . . 

1 85 
1. 140 

1. 12(1 

1 . 730 
1 . 7oo 

«. 2* 

0. 28 

«i. 2S 

<*«, In. - 








p, stmrs/eii ft 

0. INKER* 


0. 1X1238 

mi, rad iuns/ sec 


21 4 

50. 3 


4«. 8 

0. 266 

0. 748 

1. 132 


0. 720 

0. 382 

0. 362 


. ... ft. 180 

0. 225 

0. 100 


0 . ooo 

0. 143 

0. 131 

Win* station. In.*. . . . 

.... 50 




0. 375 

0. 457 

0. 417 

i 5. 56 

15. 94 

30. 88 

0. 918 


3. 63 


7. 115 


1 451 

— 0. 137 

-0. 677 

-0. 010 


0. 7H 1 


2. 72 

AC , ft 1 

7, 8tX( 

6, 080 

22. IK HI 

in. - * _ . ... . 

0. 00140 

0. 00543 

0, <KNI01 2 


•All v il'r.\< list 'd below the wine stations apply to the station indicated. 
••; here dii nfcn distunes from neutral axis to extreme fiber. 

60 x I0 4 

Airplane . 
Fuselage load 
Fuel loud 
Station, in. 

\ n r 

t’reu i»lil\ 1 j full ( ’tew <Hils 

•a full’ » 2 full Full 

/in .Mi tin 



(I>) Airplane B. 

Kicukk 12. Transfer functions. 

(>) Airplane < \ 

fa! Airplane A. 



In order to obtain an amplification or flexibility measure 
more directly comparable to the values obtained from the 
flight-test results, the following procedure was used. Equa- 
tions (58) were used to give curves of the type shown in 
sketch 1 where the ordinate X p refers to the number of 

stress peaks that occur per second above a given stress level 
represented by the abscissa. As can be seen, one curve 
applies to the flexible airplane, whereas the other is for the 
airplane considered rigid. A convenient measure of the 
magnitude of flexibility effects can be found by taking the 
ratio of the stress for the flexible case to the stress for the 
rigid case at a given value of N p (for example, the ratio of 
the stress at point 1 to the stress at point 2). In general, 
this ratio varies with stress level; it is highest at the lower 
stresses and with increasing stress decreases to a constant 
value equal to the ratio of the root-mean-square stress for 
the flexible airplane to the root-mean-square stress for the 
rigid airplane. For correlation with flight results, this ratio 
was determined for each of the three airplanes. The stress 
level chosen was in the range of the higher flight-stress values; 
specifically, it was taken equal to twice the mean-square 
stress that developed. 

Figure 13 shows a correlation of some of the results ob- 
tained by the harmonic-analysis approach with flight results. 
The ordinate is the previously used strain ratio, that is, the 
ratio of the peak incremental root bending strain for the 
flexible airplane to the peak incremental root bending strain 
for the aircraft considered rigid. The abscissa is the ratio 
obtained from the harmonic-analysis theory, as explained in 
the preceding paragraph. The three circles are the results 
for the three airplanes. As a matter of added interest, a 
single acceleration point, which was the only one computed 
and which applied to airplane B, has been inserted in the 
plot as though the coordinates involved the ratio of fuselage 
to nodal acceleration. Tin* good correlation shown l>v this 
plot is, to say the least, very gratifying; it shows tlmL good 
correlation may be obtained between calculations and flight 
results and, moreover, indicates that the harmonic-analysis 
approach is a suitable method to use. 

Figure 13. — Strain amplification for continuous turbulence. 


The derivation presented herein is intended to provide a 
convenient engineering method for taking into account wing 
bending flexibility in calculating the response of an airplane 
to either discrete or continuous-sinusoidal gusts. The 
method is believed to be well suited for making trend studies 
which evaluate, for example, the effect on response of such 
factors as mass distribution, speed, and altitude. It is not 
intended to apply for speeds near the flutter speed or for 
airplanes which have poor longitudinal damping characteris- 
tics; for these cases an extension to include wing torsion and 
airplane pitch would be desirable. 

As regards the calculations and flight studies that were 
made for three airplanes to determine the manner in which 
gust loads are magnified by wing flexibility, the following 
remarks may he made. These studies indicate that an ap- 
proach based upon single-gust encounter can he used to 
evaluate the way in which one airplane 1 compares with 
another in respect to the average of these flexibility effects. 
This discrete-gust approach also shows overall qualitative 
correlation with flight results; however, it does not permit 
detailed resolution of the flexibility effects, and lienee direct 
quantitative correlation is not feasible. A more appropriate 
approach appears to he one which considers the continuous 
random nature of atmospheric turbulence and which is based 
on generalized harmonic analysis. Not only does it permit 
airplanes to be compared with one another in detail but it 
also provides good quantitative correlation with flight 
results. It therefore appears that, through use of this 
continuous-turbulence approach, a suitable means is afforded 
for determining the magnitudes of flexibility effects. More- 
over. many useful ramifications, such as application to fatigue 
studies, are provided as well. 

Langley Aeronautical Laboratory, 

.National Advisory Committee for Aeronautics. 

Langley Field, Va., March 4. 



In this appendix, a derivation is given of equation (31) 
which gives the value of displacement in terms of successive 
past-history values of acceleration. Suppose that the sec- 
ond derivative (acceleration) of a function is approximated 
by a succession of straight-line segments as shown in sketch 2 

where the segments cover equal intervals e of the abscissa s 
and the initial condition that 2 o" = 0 is assumed to apply. 
If a dummy origin is now considered at the station m — 1 , 
the segment between stations m — 1 and m may be repre- 
sented by the equation 

Two successive integrations give the relations for z f m 
z m as follows: 


From these two equations the values of z r m and z m at any 
time interval may be given in terms of the second derivative 
at all previous time intervals. For example, with initial 
conditions of z f \— z' 0 —0, equation (Al) becomes for m= 1 

z"i (A3) 

and for m— 2 

z r 2=l U" 2 +2"l)+2'l 

Combining this equation and equation (A3) results in the 


This process may be carried through for each of the time 
stations to yield the following general equation for z' m : 

2'„=«(z".+ 2" 2 +2"3+ • ■ • +Z"»-I+! Z"~) (A4) 

which, of course, is the trapezoidal approximation of the 
area under the 2 "-curve. Equation (A2) for z m may be 
treated similarly, and it is found that the general equation 
for z m may be written 

Z„ = < j ||(w— l)2"l + (w — 2)3 "j+ . . . -f23"„_2+3"„-i+£Z"„,J 


S ^-h 2 " m 6 f-‘ S 3 + 2 ' m -.S+2„,-l 

where the constants of integration z' m . t and z m . x (initial 
conditions for the interval) have been introduced. If s is 
set equal to e in these two equations, the following equations 

z'„=\ (2".+ 2".-i)+4- I 


J2 2 

2„=g Z "™+3 Z"m-l + *'m- lt+Zm-1 


This equation thus gives the displacement at any time sta- 
tion in terms of the accelerations at all previous time 

It may be noted that, if higher-order segments (parabolic 
or cubic) had been used instead of straight-line segments to 
approximate the second derivative, equations similar in 
form to equations (A4) and (A5) would also result. For 
most practical purposes, however, the accuracy of equation 
(A5) is sufficiently good as long as the interval € is chosen so 
that the straight-line segments roughly approximate the 
second derivative. 




As a convenience, a summary of the basic steps necessary 
for calculating the response of an airplane to a discrete gust 
is given in this appendix. 

For accelerations and displacements: 

(1) With the use of the fundamental mode, wing plan 
form, and mass distribution, calculate the quantities 

X, r u and r 3 as given by equations (18). 

(2) Choose the time interval «. A convenient rule of 

thumb is but for most cases «=1 should give satis- 


factory results. 

(3) Determine values of the unsteady-lift function 0= 1 -<t> 
at successive multiple intervals of «. (See fig. 1.) Also 
determine corresponding values of the gust-force integral 
f(s), equation (27b). As an aid, curves for f(a) are presented 
in figure 1 for the sharp-edge gust and in figure 14 for various- 
length sine gusts, sine 2 gusts, and triangular gusts. (The 

curves in fig. 1 have been obtained from eqs. (9) and (10). 
These approximations, although rather accurate for the 
lower values of s , are noted to cross; actually, they should 
not cross and are known to have the same asymptotic 
approach to unity.) 

(4) From the following definitions: 

Ai — Mo+ «0Q 






Bi — rjtflo 



C m ={m— 1) ^ t 2 X J +2 (“'■i) «#*-t 

(m> 1) 

Imcckk 14. — Value of the guM -force integral /(*) = p — o)*la for three gust shapes. 




Fiocre 14. — ('onritmcri. 

set up the following matrices: 


[-11 = 

At . 1 , 

. 1 , - 1 , - 1 . 

. 1 4 At . 1 . . 1 , 



r, (\ 

( 3 r, <\ 

<\ ('■ <\ 

h, n, 







Then, ealeulate the matrix 



(5) Solve for tin* values of $ (which e<|tials from 

equation (33) bv the method outlined after equation (33). 
(See eq. (34).) The values of and a (whicli equals ) 
can then he calculated from equations (31) and (32). 

For bending moment: 

(6) In order to compute bending moment, determine r 3 . 
tj 0 , and tj, as given by equations (24), where M mn . M mi , \f,, y 



(c) //= 7.") chords. 
Fiia’RK 14. — ( 'outiimed. 

and M e in those equations depend on the particular will" 
station being considered and are given by equations (21). 

(7) Determine bending moment by use of equations (2B) 
with the values of response already established. This 
equation may be applied directly to any desired time value. 
Maximum bending moment usually occurs very close to the 
time when Z\ is a maximum. 


1 Donely, Philip: Summary of Information Relating to Gust Loads 
on Airplanes. XACA Rep. 997 , 1950 . (Supersedes XACA 
TX 1976.) 

2. Bisplinghoff, K. L., Isakson, G., and O’Brien, T. F.: Oust Loads 
on Rigid Airplanes With Pitching Neglected. Jour. Aero. Sei., 
vol. 18, no. 1, Jan. 1951, pp. 88-42. 

Risplinghoff, It. I.„ Isakson, C.. f and O’Brien, T. F.: Report on 
( 1 1 ist Loads on Rigid arid Elastic Airplanes. Contract No. 
XOa(s) 8790, M.I.T. Rep., Bur. Aero.. Aug. 15, 1949. ( This 
citation is intended to include ref. 8 of this reference paper which 
is listed therein as “Greidanus, J. H.: ‘The Loading of Airplane 
Structures by Symmetrical Gusts’ (in Dutch with an English 
summary). Xationaal Luchtvaartlaboratorium, Amsterdam, 
Rapport No. XIV, 1948.") 

4 Ma/.olsky Bernard: (’harts of Airplane Acceleration Ratio for 
Gusts *«f Arbitrary Shape. XACA TX 2080, 1950. 

5. Greidanus, J. H., and Van de Vooren, A. T.: Gust Load ( ’oefficiunt> 
for Wing and Tail Surfaces of an Aeroplane. Rep. F.28. 
Xationaal Luchtvaartlaboratorium, Amsterdam. Dec. 1948. 
t>. Press, Harry, and Mazelskv, Bernard: A Study of the Application 
of Power-Spectral Methods of Generalized Harmonic Annlyd* 
to Gust Loads on Airplanes. XACA TX 2858. 1958. 

7. Clementson, Gerhardt ('.: An Investigation of the Power Spectral 

Density of Atmospheric Turbulence. Ph D Thesis, M.I.T . 195(1. 

8. Liepmaim, H. W.: On the Application of Statistical Concept* to 

the Buffeting Problem. Jour. Aero. Sei., vol. 19, no. 12. Dec. 
1952, pp. 798-800, 822. 

9. Fung, V. C.: Statistical Aspects of Dynamic Loads. Jour. Aero. 

Sci., vol. 21), no. 5, May 1958, pp. 817 -880. 

10. Houbolt, John (\: Correlation of Calculation and Flight Studies 

of the Effect of Wing Flexibility on Structural Response Due to 
Gusts. XACA TX 8006, 1958. 

11. Houbolt, John (\, and Kordcs, Eldon E.: Gust-Response Analysis 

of an Airplane Including Wing Bending Flexibility. XACA 
TX 2768, 1952. 

12. Kordes, Eldon K., and Houbolt, John ( .: Evaluation of Gust 

Response Characteristics of Some Existing Aircraft With Wing 
Bending Flexibility Included. XACA TX 2897, 1958. 

18. Bisplinghoff, R. L., Isakson, G. f Pian, T. H H., Floinenhoft . H I., 
and O’Brien, T. F.: An Investigation of Stresses in Aircraft 
Structures Cnder Dynamic Loading. Contract No. XOa(>' 
8790, M. I. T. Rep., Bur Aero., Jan. 21, 1949. 



(d) H= 10 chords. 
Figure 14. — Concluded. 

14. Goland, M. f Luke, Y. L., and Kahn, E. A.: Prediction of Wing 
Loads Due to Gusts Including Aero- Elastic Effects. Part I — 
Formulation of the Method. AF TR No. 5706, Air Materiel 
Command, U. S. Air Force, July 21, 1947. 

15* Radok, J. R. M., and Stiles, Lurline F.: The Motion and Deforma- 
tion of Aircraft in Uniform and Non-Uniform Atmospheric Dis- 
turbances. Rep. ACA-41, Council for Sci. and Ind. Res., Div. 
Aero., Commonwealth of Australia, 1948. 

16. Houbolt, John C.: A Recurrence Matrix Solution for the Dynamic 

Response of Aircraft in Gusts. NACA Rep. 101C, 1951. (Super- 
sedes NACA TN 2060.) 

17. Jones, Robert T.: The Unsteady Lift of a Wing of Finite Aspect 

Ratio. NACA Rep. 681, 1940. 

18. Houbolt, John C\, and Anderson, Roger A.: Calculation of Un- 

coupled Modes and Frequencies in Bending or Torsion of Non- 
uniform Beams. NACA TN 1522, 1948. 

19. Radok, J. R. M.: The Problem of Gust Loads on Aircraft. A 

Survey of the Theoretical Treatment. Rep SM. 133, Dept. 
Supply and Dev., Div Aero., Commonwealth of Australia, 
July 1949. 

20. Biot, M. A.: Loads on a Supersonic W r ing Striking a Sharp- Edged 

Gust. Jour. Aero. Sci., vol. 16, no. 5, May 1949, pp, 296-300, 

21. Miles, John W.: Transient Loading of Supersonic Rectangular 

Airfoils. Jour. Aero. Sci., vol. 17, no. 10, Oct 1950, pp. 647-652. 

22. Miles, John W.; The Indicial Admittance of a Supersonic Rectan- 

gular Airfoil. NAVORD Rep. 1171, U. S. Naval Ord Test 
Station (Inyokern, Calif.), July 21, 1949. 

23. James, Hubert M., Nichols, Nathaniel B., and Phillips, Ralph S.: 

Theory of Servomechanisms. McGraw-Hill Book C'o., Inc., 
1947, pp. 202-307. 

24. Rice, S. O. : Mathematical Analysis of Random Noise. Pts. I and 

II. Bell Syst. Tech. Jour., vol. XXIII, no. 3, July 194-1, pp. 
282-332; Pts. Ill and IV, vol. XXIV, no. 1, Jan. 1945, pp. 46-156. 

25. Shufflebarger, C. C., and Mickleboro, Harry C\: Flight Investiga- 

tion of the Effect of Transient Wing Response on Measured 
Accelerations of a Modern Transport Airplane in Rough Air 
NACA TN 2150, 1950. 

26. Mickleboro, Harry C., and Shufflebarger, (\ (\: Flight Investiga- 

tion of the Effect of Transient Wing Response on Wing Strains 
of a Twin-Engine Transport Airplane in Rough Air. NACA 
TN 2424, 1951. 

27. Mickleboro, Harry Fahrer, Richard B., and Shufflebarger. 

C. C.: Flight Investigation of Transient Wing Response on a 
Four-Engine Bomber Airplane in Rough Air With Respect to 
Center-of-Gravity Accelerations. NACA TN 2780, 1952 

28. Murrow, HaroUJ N., and Payne, Chester B.: Flight Investigation 

of the Effect of Transient Wing Response on Wing Strains of a 
Four-Engine Bomber Airplane in Rough Air. NACA TN 
2951, 1953.